Computational Geomechanics. Manuel Pastor

Computational Geomechanics

us now consider the effect of the simultaneous application of a total external hydrostatic stress and a pore pressure change, both equal to Δp, to any porous material. The above requirement can be written in tensorial notation as requiring that the total stress increment is defined as

(1.8a) normal upper Delta sigma Subscript italic i j Baseline equals minus delta Subscript italic i j Baseline normal upper Delta p

or, using the vector notation

(1.8b) normal upper Delta bold sigma equals minus bold m normal upper Delta p

In the above, the negative sign is introduced since “pressures” are generally defined as being positive in compression, while it is convenient to define stress components as positive in tension.

It is evident that for the loading mentioned, only a very uniform and small volumetric strain will occur in the skeleton and the material will not suffer any damage provided that the grains of the solid are all made of identical material. This is simply because all parts of the porous medium solid component will be subjected to identical compressive stress.

However, if the microstructure of the porous medium is composed of different materials, it appears possible that nonuniform, localized stresses, can occur and that local grain damage may be suffered. Experiments performed on many soils and rocks and rock‐like materials show, however, that such effects are insignificant. Thus, in general, the grains and, hence, the total material will be in a state of pure volumetric strain

(1.9) normal upper Delta epsilon Subscript v Baseline almost-equals normal upper Delta epsilon Subscript italic i i Baseline equals normal upper Delta epsilon 11 plus normal upper Delta epsilon 22 plus normal upper Delta epsilon 33 equals minus StartFraction 1 Over upper K Subscript s Baseline EndFraction normal upper Delta p

where K_s is the average material bulk modulus of the solid components of the skeleton. Alternatively, adopting a vectorial notation for strain in a manner involved in (1.1)

(1.10a) normal upper Delta epsilon Subscript v Baseline equals bold m Superscript normal upper T Baseline normal upper Delta bold epsilon equals minus StartFraction 1 Over upper K Subscript s Baseline EndFraction normal upper Delta p

where ε is the vector defining the strains in the manner corresponding to that of stress increment definition. Again, assuming that the material is isotropic, we shall have

(1.10b) normal upper Delta bold epsilon equals minus bold m StartFraction 1 Over 3 upper K Subscript s Baseline EndFraction normal upper Delta p

Those not familiar with soil mechanics may find the following hypothetical experiment illustrative. A block of porous, sponge‐like rubber is immersed in a fluid to which an increase in pressure of Δp is applied as shown in Figure 1.4. If the pores are connected to the fluid, the volumetric strain will be negligible as the solid components of the sponge rubber are virtually incompressible.

If, on the other hand, the block is first encased in a membrane and the interior is allowed to drain freely, then again a purely volumetric strain will be realized but now of a much larger magnitude.

The facts mentioned above were established by the very early experiments of Fillunger (1915) and it is surprising that so much discussion of “area coefficients” has since been necessary.

From the preceding discussion, it is clear that if the material is subject to a simultaneous change of total stress Δσ and of the total pore pressure Δp, the resulting strain can always be written incrementally in tensorial notation as

(1.11a)

or in vectoral notation

(1.11b) normal upper Delta bold epsilon equals bold upper D Superscript negative 1 Baseline left-parenthesis normal upper Delta bold sigma plus bold m normal upper Delta p right-parenthesis minus bold m StartFraction 1 Over 3 upper K Subscript s Baseline EndFraction normal upper Delta p plus normal upper Delta bold epsilon Superscript 0

with

(11.11c) upper C Subscript italic ijkl Baseline upper D Subscript italic mnop Baseline equals delta Subscript italic im Baseline delta Subscript italic j n Baseline delta Subscript italic k o Baseline delta Subscript italic l p

Schematic illustration of a porous material subject to external hydrostatic pressure increases delta p, and (a) internal pressure increment delta p; (b) internal pressure increment of zero.

Figure 1.4 A porous material subject to external hydrostatic pressure increases Δ_p, and (a) internal pressure increment Δ_p; (b) internal pressure increment of zero.

The last term in (1.11a) and (1.11b), Δε⁰, is simply the increment of an initial strain such as may be caused by temperature changes, etc., while the penultimate term is the strain due to the grain compression already mentioned, viz. Equation (1.10). D is a tangent matrix of the solid skeleton implied by the constitutive relation with corresponding compliance coefficient matrix D ⁻¹ = C. These, of course, could be matrices of constants, if linear elastic behavior is assumed, but generally will be defined by an appropriate nonlinear relationship of the type which we shall discuss in Chapter 4 and this behavior can be established by fully drained (p = 0) tests.

Although the effects of skeleton deformation due to the effective stress defined by (1.6) with n_w = 1 have been simply added to the uniform volumetric compression, the principle of superposition requiring linear behavior is not invoked and in this book, we shall almost exclusively be concerned with nonlinear, irreversible, elastoplastic and elastoviscoplastic responses of the skeleton which, however, we assume incremental properties.

For assessment of the strength of the saturated material, the effective stress previously defined with n_w = 1 is sufficient. However, we note that the deformation relation of (1.11) can always be rewritten incorporating the small compressive deformation of the particles as (1.12).

It is more logical at this step to replace the finite increment by an infinitesimal one and to invert the relations in (1.11) writing these as

(1.12a)Скачать книгу